ode_rk45_step

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ode_rk45_step#

class braintools.quad.ode_rk45_step(f, y, t, *args, return_error=False, **kwargs)#

One step of the Cash-Karp embedded Runge-Kutta 4(5) method.

Computes a 5th-order solution and a 4th-order embedded solution using six stages. Optionally returns a PyTree error estimate y5 - y4 for adaptive step-size controllers.

Parameters:

  • f (Callable[[PyTree, float | Quantity, ...], PyTree]) – Right-hand side function f(y, t, *args) -> PyTree.

  • y (PyTree) – Current state at time t.

  • t (Array | ndarray | bool | number | bool | int | float | complex | Quantity) – Current time.

  • *args – Additional positional arguments forwarded to f.

  • return_error (bool) – If True, also return a PyTree error estimate (y5 - y4).

Returns:

The updated state (5th-order). If return_error is True, returns (y_next, error_estimate) where both are PyTrees matching y.

Return type:

PyTree or tuple

Notes

Butcher tableau (c, a, b5, b4): - c = [0, 1/5, 3/10, 3/5, 1, 7/8] - a21 = 1/5 - a31 = 3/40, a32 = 9/40 - a41 = 3/10, a42 = -9/10, a43 = 6/5 - a51 = -11/54, a52 = 5/2, a53 = -70/27, a54 = 35/27 - a61 = 1631/55296, a62 = 175/512, a63 = 575/13824, a64 = 44275/110592, a65 = 253/4096 - b5 = [37/378, 0, 250/621, 125/594, 0, 512/1771] - b4 = [2825/27648, 0, 18575/48384, 13525/55296, 277/14336, 1/4]

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